留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

一类KdV-Burgers方程的奇摄动解与孤子解

包立平 李瑞翔 吴立群

包立平, 李瑞翔, 吴立群. 一类KdV-Burgers方程的奇摄动解与孤子解[J]. 应用数学和力学, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011
引用本文: 包立平, 李瑞翔, 吴立群. 一类KdV-Burgers方程的奇摄动解与孤子解[J]. 应用数学和力学, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011
BAO Liping, LI Ruixiang, WU Liqun. Singularly Perturbed and Soliton Solutions to a Class of KdV-Burgers Equations[J]. Applied Mathematics and Mechanics, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011
Citation: BAO Liping, LI Ruixiang, WU Liqun. Singularly Perturbed and Soliton Solutions to a Class of KdV-Burgers Equations[J]. Applied Mathematics and Mechanics, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011

一类KdV-Burgers方程的奇摄动解与孤子解

doi: 10.21656/1000-0887.420011
基金项目: 

国家自然科学基金(51775154);浙江省重点自然科学基金(LZ15E050004)

详细信息
    作者简介:

    包立平(1962—),男,副教授,博士(E-mail: baolp@hdu.edu.cn);李瑞翔(1996—),男,硕士生(通讯作者. E-mail: 843271281@qq.com).

    通讯作者:

    李瑞翔(1996—),男,硕士生(通讯作者. E-mail: 843271281@qq.com).

  • 中图分类号: O175.29

Singularly Perturbed and Soliton Solutions to a Class of KdV-Burgers Equations

Funds: 

The National Natural Science Foundation of China(51775154)

  • 摘要: 讨论了一类具有大Reynolds数且弱频散性的KdV-Burgers方程, 在数学上表示为一类奇摄动KdV-Burgers方程.KdV-Burgers方程中含有的非线性项与频散项互补作用形成稳定向前传播的孤立子.通过数学分析, 描述了孤立子的传播途径和传播速度等物理量的发展变化规律.通过奇摄动展开方法, 构造了该问题的渐近解.首先,用Riemann-Earnshaw方法求得退化解, 得到了简单波, 该简单波波形中的任意一点与初始点都存在一个传播速度差, 这使得波在传播过程中波形不断畸变, 最终形成冲击波面, 即间断面, 在它的两侧质点的速度有一个跳跃, 且随时间不断变化;其次, 在退化解的间断曲面处做变量替换, 构造一种修正的行波变换, 得到了内解展开式的孤子解, 并证明了内外解的存在性与唯一性;最后,通过一致有界逆算子的存在性做了余项估计, 并得到渐近解的一致有效性.结果表明, KdV-Burgers方程在大Reynolds数且弱频散性的性质下,扰动集中在退化解的间断面附近,孤立子链接两侧质点,其传播途径不是时间与空间的线性形式,而是沿着退化解的间断面附近传播,形成稳定的波形.
  • 王建勇, 程雪苹, 曾莹, 等. Korteweg-de Vries方程的准孤立子解及其在离子声波中的应用[J]. 物理学报,2018,67(11): 110201.(WANG Jianyong, CHENG Xueping, ZENG Ying, et al. Quasi-soliton solution of Korteweg-de Vries equation and its application in ion acoustic waves[J].Acta Physica Sinica,2018,67(11): 110201.(in Chinese))
    [2]王建勇. KdV方程的孤子-椭圆周期波解及其准孤立子行为[J]. 宁波大学学报(理工版), 2020,33(5): 62-67.(WANG Jianyong. Soliton-cnoidal wave solution and its quasi-soliton behavior to the Korteweg-de Vries equation[J].Journal of Ningbo University(Natural Science & Engineering Edition),2020,33(5): 62-67.(in Chinese))
    [3]KUDRYASHOV N A. Traveling wave reduction of the modified KdV hierarchy: the lax pair and the first integrals[J].Communications in Nonlinear Science and Numerical Simulation,2019,73: 472-480.
    [4]王慧敏. (3+1)维修正KdV-Zakharov-Kuznetsov方程孤波的格子Boltzmann模拟[J].数码世界, 2020(10): 239-240.(WANG Huiming. The lattice Boltzmann simulation of solitary waves for the (3+1) dimensional modified KdV-Zakharov-Kuznetsov equation[J].Digital Space,2020(10): 239-240.(in Chinese))
    [5]ZHANG G Q, YAN Z Y. Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions[J].Physica D: Nonlinear Phenomena,2020,402: 132170.
    [6]DUBROVIN B, YANG D, ZAGIER D. On tau-functions for the KdV hierarchy[J].Selecta Mathematica,2021,27(1). DOI: 10.1007/s00029-021-00620-x.
    [7]CHEN K, DENG X, LOU S. Solutions of nonlocal equations reduced from the AKNS hierarchy[J].Studies in Applied Mathematics,2018,141(1): 113-141.
    [8]BURYAK A. Open intersection numbers and the wave function of the KdV hierarchy[J].Moscow Mathematical Journal,2016,16 (1): 27-44.
    [9]IGNATYEV M Y. On the solutions of some boundary value problems for the general KdV equation[J].Mathematical Physics, Analysis and Geometry,2014,17: 493-509.
    [10]DAI C Q, WANG Y Y, FAN Y, et al. Interactions between exotic multi-valued solitons of the (2+1)-dimensional Korteweg-de Vries equation describing shallow water wave[J].Applied Mathematical Modelling,2020,80: 506-515.
    [11]KUDRYASHOV N A. Solitary wave solutions of hierarchy with non-local nonlinearity[J].Applied Mathematics Letters,2020,103: 106155.
    [12]ZHUANG K G, DU Z J, LIN X J. Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method[J].Nonlinear Dynamics,2015,80: 629-635.
    [13]XU Y, DU Z J, WEI L. Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation[J].Nonlinear Dynamics,2016,83: 65-73.
    [14]CHEN S S, TIAN B, LIU L, et al. Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system[J].Chaos, Solitons & Fractals,2019,118: 337-346.
    [15]BASKONUS H M, SULAIMAN T A, BULUT H, et al. Investigations of dark, bright, combined dark-bright optical and other soliton solutions in the complex cubic nonlinear Schrödinger equation with δ-potential[J].Superlattices and Microstructures,2018,115: 19-29.
    [16]ESEN A, SULAIMAN T A, BULUT H, et al. Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation[J].Optik,2018,167: 150-156.
    [17]PRIYA N V, SENTHILVELAN M. N-bright-bright and N-dark-dark solitons of the coupled generalized nonlinear Schrödinger equations[J].Communications in Nonlinear Science and Numerical Simulation,2016,36: 366-377.
    [18]BENDAHMANE I, TRIKI H, BISWAS A, et al. Bright, dark and W-shaped solitons with extended nonlinear Schrödinger’s equation for odd and even higher-order terms[J].Superlattices and Microstructures,2018,114: 53-61.
    [19]KUMAR S, NIWAS M, HAMID I. Lie symmetry analysis for obtaining exact soliton solutions of generalized Camassa-Holm-Kadomtsev-Petviashvili equation[J].International Journal of Modern Physics B,2021,35(2): 2150028.
    [20]BEKIR A, ZAHRAN E H M, GUNER O. Soliton solutions of the (3+1)-dimensional Yu-Toda-Sassa-Fukuyama equation by the new approach and its numerical solutions[J].International Journal of Modern Physics B,2021,35(2): 2150025.
    [21]FU H M, RUAN C Z, HU W Y. Soliton solutions to the nonlocal Davey-Stewartson Ⅲ equation[J].Nature Communications,2021,35(1): 2150026.
    [22]钱祖文. 非线性声学[M]. 2版. 北京: 科学出版社, 2009.(QIAN Zuwen.Non-Linear Acoustics[M]. 2nd ed. Beijing: Science Press, 2009.(in Chinese))
    [23]叶彦谦. 常微分方程讲义[M]. 北京: 人民教育出版社, 1979.(YE Yanqian.Handout for Ordinary Differential Equations[M]. Beijing: People’s Education Press, 1979.(in Chinese))
  • 加载中
计量
  • 文章访问数:  71
  • HTML全文浏览量:  18
  • PDF下载量:  12
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-01-21
  • 修回日期:  2021-03-24

目录

    /

    返回文章
    返回